The mathematical foundations of general relativity revisited

The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in order to revisit the mathematical foundations of general relativity. Other engineering examples (control theory, elasticity theory, electromagnetism) will also be considered in order to illustrate the three fundamental results that we shall provide. The paper is therefore divided into three parts corresponding to the different formal methods used. 1) CARTAN VERSUS VESSIOT: The quadratic terms appearing in the"Riemann tensor"according to the"Vessiot structure equations"must not be identified with the quadratic terms appearing in the well known"Cartan structure equations"for Lie groups and a similar comment can be done for the"Weyl tensor". In particular,"curvature+torsion"(Cartan) must not be considered as a generalization of"curvature alone"(Vessiot). Roughly, Cartan and followers have not been able to"quotient down to the base manifold", a result only obtained by Spencer in 1970 through the"nonlinear Spencer sequence"but in a way quite different from the one followed by Vessiot in 1903 for the same purpose and still ignored. 2) JANET VERSUS SPENCER: The"Ricci tensor"only depends on the nonlinear transformations (called"elations"by Cartan in 1922) that describe the"difference"existing between the Weyl group (10 parameters of the Poincar\'e subgroup + 1 dilatation) and the conformal group of space-time (15 parameters). It can be defined by a canonical splitting, that is to say without using the indices leading to the standard contraction or trace of the Riemann tensor. Meanwhile, we shall obtain the number of components of the Riemann and Weyl tensors without any combinatoric argument on the exchange of indices. Accordingly, the Spencer sequence for the conformal Killing system and its formal adjoint fully describe the Cosserat/Maxwell/Weyl theory but General Relativity is not coherent at all with this result. 3) ALGEBRAIC ANALYSIS: Contrary to other equations of physics (Cauchy equations, Cosserat equations, Maxwell equations), the Einstein equations cannot be"parametrized", that is the generic solution cannot be expressed by means of the derivatives of a certain number of arbitrary potential-like functions, solving therefore negatively a 1000 $ challenge proposed by J. Wheeler in 1970. Accordingly, the mathematical foundations of mathematical physics must be revisited within this formal framework, though striking it may look like for certain apparently well established theories such as electromagnetism and general relativity. We insist on the fact that the arguments presented are of a purely mathematical nature and are thus unavoidable.

algebraic analogue of a differential sequence. Finally, we get in the third part of the paper: THIRD FUNDAMENTAL RESULT: Contrary to other equations of physics (Cauchy equations, Cosserat equations, Maxwell equations), the Einstein equations cannot be " parametrized ", that is the generic solution cannot be expressed by means of the derivatives of a certain number of arbitrary potential-like functions, solving therefore negatively a 1000 $ challenge proposed by J. Wheeler in 1970.
As no one of these results can be obtained without the previous difficult purely mathematical arguments and are thus unavoidable, the purpose of this paper is to present them for the first time in a rather self-contained and elementary way through explicit basic examples.

INTRODUCTION:
The purpose of this paper is to present an elementary summary of a few recent results obtained through the application of the formal theory of systems of ordinary differential (OD) or partial differential (PD) equations and Lie pseudogroups in order to revisit the mathematical foundations of General relativity (GR). More elementary engineering examples (elasticity theory, electromagnetism (EM)) will also be considered in order to illustrate the quoted three fundamental results that we shall provide. The paper, based on the material of two lectures given at the department of mathematics of the university of Montpellier 2, France, in may 2013, is divided into three parts corresponding to the different formal methods used. 1) FIRST PART: Lie groups of transformations may be considered as Lie pseudogroups of transformations, that is to say groups of transformations solutions of systems of OD or PD equations, but no action type method can be used as parameters never appear any longer.
2) SECOND PART: The work of Cartan is superseded by the use of the canonical Spencer sequence while the work of Vessiot is superseded by the use of the canonical Janet sequence but the link between these two sequences and thus these two works is not known today.
3) THIRD PART: Using duality theory, the formal adjoint of the Spencer operator for the conformal group of transformations of space-time provides the Cosserat equations, the Maxwell equations and the Weyl equations on equal footing but such a result, even if it allows to unify the finite elements of engineering sciences, also leads to contradictions in GR that we shall point out.
The new methods involve tools from differential geometry (jet theory, Spencer operator, δcohomology) and homological algebra (diagram chasing, snake theorem, extension modules, double duality). The reader may just have a look to the book ( [18], review in Zbl 1079.93001) in order to understand the amount of mathematics needed from many domains.
The following diagram summarizes at the same time the historical background and the difficulties presented in the abstract: Roughly, Cartan and followers have not been able to " quotient down to the base manifold " ( [1,2]), a result only obtained by Spencer in 1970 through the nonlinear Spencer sequence ( [5], [9], [15], [22]) but in a way quite different from the one followed by Vessiot in 1903 for the same purpose ( [17], [25]). Accordingly, the mathematical foundations of mathematical physics must be revisited within this formal framework, though striking it may look like for certain apparently well established theories such as EM and GR.

FIRST PART : FROM LIE GROUPS TO LIE PSEUDOGROUPS
If X is a manifold with local coordinates (x i ) for i = 1, ..., n = dim(X), let E be a fibered manifold over X, that is a manifold with local coordinates (x i , y k ) for i = 1, ..., n and k = 1, ..., m simply denoted by (x, y), projection π : E → X : (x, y) → (x) and changes of local coordinates x = ϕ(x),ȳ = ψ(x, y). If E and F are two fibered manifolds over X with respective local coordinates (x, y) and (x, z), we denote by E× X F the fibered product of E and F over X as the new fibered manifold over X with local coordinates (x, y, z). We denote by f : X → E : (x) → (x, y = f (x)) a global section of E, that is a map such that π • f = id X but local sections over an open set U ⊂ X may also be considered when needed. We shall use for simplicity the same notation for a fibered manifold and its set of sections while setting dim X (E) = m. Under a change of coordinates, a section transforms likef (ϕ(x)) = ψ(x, f (x)) and the derivatives transform like: We may introduce new coordinates (x i , y k , y k i ) transforming like: ) where both f q and j q (f ) are over the section f of E. Of course J q (E) is a fibered manifold over X with projection π q while J q+r (E) is a fibered manifold over J q (E) with projection π q+r q , ∀r ≥ 0. DEFINITION 1.1: A (nonlinear) system of order q on E is a fibered submanifold R q ⊂ J q (E) and a solution of R q is a section f of E such that j q (f ) is a section of R q . DEFINITION 1.2: When the changes of coordinates have the linear formx = ϕ(x),ȳ = A(x)y, we say that E is a vector bundle over X. Vector bundles will be denoted by capital letters C, E, F and will have sections denoted by ξ, η, ζ. In particular, we shall denote as usual by T = T (X) the tangent bundle of X, by T * = T * (X) the cotangent bundle, by ∧ r T * the bundle of r-forms and by S q T * the bundle of q-symmetric tensors. When the changes of coordinates have the form x = ϕ(x),ȳ = A(x)y + B(x) we say that E is an affine bundle over X and we define the associated vector bundle E over X by the local coordinates ( Finally, If E = X × X, we shall denote by Π q = Π q (X, X) the open subfibered manifold of J q (X × X) defined independently of the coordinate system by det(y k i ) = 0 with source projection α q : Π q → X : (x, y q ) → (x) and target projection β q : Π q → X : (x, y q ) → (y).
we may introduce the vertical bundle V (E) ⊂ T (E) as a vector bundle over E with local coordinates (x, y, v) obtained by setting u = 0 and changes v l = ∂ψ l ∂y k (x, y)v k . Of course, when E is an affine bundle over X with associated vector bundle E over X, we have V (E) = E × X E.
For a later use, if E is a fibered manifold over X and f is a section of E, we denote by f −1 (V (E)) the reciprocal image of V (E) by f as the vector bundle over X obtained when replacing (x, y, v) by (x, f (x), v) in each chart. A similar construction may also be done for any affine bundle over E.
We now recall a few basic geometric concepts that will be constantly used through this paper. First of all, if ξ, η ∈ T , we define their bracket [ξ, η] ∈ T by the local formula ([ξ, η] When I = {i 1 < ... < i r } is a multi-index, we may set dx I = dx i1 ∧ ... ∧ dx ir for describing ∧ r T * by means of a basis and introduce the exterior derivative d : The Lie derivative of an r-form with respect to a vector field ξ ∈ T is the linear first order operator L(ξ) linearly depending on j 1 (ξ) and uniquely defined by the following three properties: We now turn to group theory and start with two basic definitions: Let G be a Lie group, that is a manifold with local coordinates (a τ ) for τ = 1, ..., p = dim(G) called parameters, a composition G × G → G : (a, b) → ab, an inverse G → G : a → a −1 and an identity e ∈ G satisfying: (ab)c = a(bc) = abc, aa −1 = a −1 a = e, ae = ea = a, ∀a, b, c ∈ G DEFINITION 1.4: G is said to act on X if there is a map X ×G → X : (x, a) → y = ax = f (x, a) such that (ab)x = a(bx) = abx, ∀a, b ∈ G, ∀x ∈ X and we shall say that we have a Lie group of transformations of X. In order to simplify the notations, we shall use global notations even if only local actions are existing. It is well known that the action of G onto itself allows to introduce a purely algebraic bracket on its Lie algebra G = T e (G).

DEFINITION 1.5:
A Lie pseudogroup of transformations Γ ⊂ aut(X) is a group of transformations solutions of a system of OD or PD equations such that, if y = f (x) and z = g(y) are two solutions, called finite transformations, that can be composed, then z = g • f (x) = h(x) and x = f −1 (y) = g(y) are also solutions while y = x is the identity solution denoted by id = id X and we shall set id q = j q (id). In all the sequel we shall suppose that Γ is transitive that is ∀x, y ∈ X, ∃f ∈ Γ, y = f (x) It becomes clear that Lie groups of transformations are particular cases of Lie pseudogroups of transformations as the system defining the finite transformations can be obtained by eliminating the parameters among the equations y q = j q (f )(x, a) when q is large enough. The underlying system may be nonlinear and of high order. Looking for transformations "close" to the identity, that is setting y = x + tξ(x) + ... when t ≪ 1 is a small constant parameter and passing to the limit t → 0, we may linearize the above nonlinear system of finite Lie equations in order to obtain a linear system of infinitesimal Lie equations of the same order for vector fields. Such a system has the property that, if ξ, η are two solutions, then [ξ, η] is also a solution. Accordingly, the set Θ ⊂ T of solutions of this new system satisfies [Θ, Θ] ⊂ Θ and can therefore be considered as the Lie algebra of Γ. EXAMPLE 1.6: While the affine transformations y = ax + b are solutions of the second order linear system y xx = 0, the projective transformations y = (ax + b)/(cx + d) are solutions of the third order nonlinear system Ψ ≡ (y xxx /y x ) − 3 2 (y xx /y x ) 2 = 0. The sections of the corresponding linearized systems are respectively satisfying ξ xx = 0 and ξ xxx = 0. The generating differential invariant Φ ≡ y xx /y x of the affine case transforms like u =ū∂ x f + (∂ xx f /∂ x f ) whenx = f (x) and we let the reader exhibit the corresponding change for Ψ as an exercise.
We now sketch the discovery of Vessiot ([17], [25]) still not known today after more than a century for reasons which are not scientific at all. Roughly, a Lie pseudogroup Γ ⊂ aut(X) is made by finite transformations y = f (x) solutions of a (possibly nonlinear) system R q ⊂ Π q while the infinitesimal transformations ξ ∈ Θ are solutions of the linearized system R q = id −1 q (V (R q )) ⊂ J q (T ) as we have T = id −1 (V (X × X). When Γ is transitive, there is a canonical epimorphism π q 0 : R q → T . Also, as changes of source x commute with changes of target y, they exchange between themselves any generating set of differential invariants {Φ τ (y q )} as in the previous example.Then one can introduce a natural bundle F over X, also called bundle of geomeric objects, by patching changes of coordinates of the formx = f (x),ū = λ(u, j q (f (x)) thus obtained. A section ω of F is called a geometric object or structure on X and transforms likeω( . This is a way to generalize vectors and tensors (q = 1) or even connections (q = 2). As a byproduct we have Γ = {f ∈ aut(X)|j q (f ) −1 (ω) = ω} and we may say that Γ preserves ω. Replacing j q (f ) by f q , we also obtain R q = {f q ∈ Π q |f −1 q (ω) = ω}. Coming back to the infinitesimal point of view and setting f t = exp(tξ) ∈ aut(X), ∀ξ ∈ T , we may define the ordinary Lie derivative with value in the vector bundle F 0 = ω −1 (V (F )) by the formula : and we say that D is a Lie operator because Dξ = 0, Dη = 0 ⇒ D[ξ, η] = 0 as we already saw.
Differentiating r times the equations of R q that only depend on j 1 (ω), we may obtain the r-prolongation R q+r = J r (R q ) ∩ J q+r (T ) ⊂ J r (J q (T )). The problem is then to know under what conditions on ω all the equations of order q + r are obtained by r prolongations only, ∀r ≥ 0 or, equivalently, R q is formally integrable (FI). The solution, found by Vessiot, has been to exhibit another natural vector bundle F 1 with local coordinates (x, u, v) over F with local coordinates (x, u) and to prove that an equivariant section c : F → F 1 : (x, u) → (x, u, v = c(u)) only depends on a finite number of constants called structure constants. The integrability conditions (IC) of R q , called Vessiot structure equations, are of the form I(j 1 (ω)) = c(ω) and are invariant under any change of source.
We provide in a self-contained way and parallel manners the following five striking examples which are among the best nontrivial ones we know and invite the reader to imagine at this stage any possible link that could exist between them (A few specific definitions will be given later on). EXAMPLE 1.7: Coming back to the last example, we show that Vessiot structure equations may even exist when n = 1. For this, if γ is the geometric object of the affine group y = ax + b and 0 = α = α(x)dx ∈ T * is a 1-form, we consider the object ω = (α, γ) and get at once the two second order Medolaghi equations: Differentiating the first equation and substituting the second, we get the zero order equation: EXAMPLE 1.8: (Principal homogeneous structure) When Γ is the Lie group of transformations made by the constant translations y i = x i + a i for i = 1, ..., n of a manifold X with dim(X) = n, the characteristic object invariant by Γ is a family ω = (ω τ = ω τ i dx i ) ∈ T * × X ... × X T * of n 1-forms with det(ω) = 0 in such a way that Γ = {f ∈ aut(X)|j 1 (f ) −1 (ω) = ω} where aut(X) denotes the pseudogroup of local diffeomorphisms of X, j q (f ) denotes the derivatives of f up to order q and j 1 (f ) acts in the usual way on covariant tensors. For any vector field ξ ∈ T = T (X) the tangent bundle to X, introducing the standard Lie derivative L(ξ) of forms with respect to ξ, we may consider the n 2 first order Medolaghi equations: The particular situation is found with the special choice ω = (dx i ) that leads to the involutive system ∂ i ξ k = 0. Introducing the inverse matrix α = (α i τ ) = ω −1 , the above equations amount to the bracket relations [ξ, α τ ] = 0 and, using crossed derivatives on the solved form , we obtain the n 2 (n − 1)/2 zero order equations: The integrability conditions (IC), that is the conditions under which these equations do not bring new equations, are thus the n 2 (n − 1)/2 Vessiot structure equations: . When X = G, these equations can be identified with the Maurer-Cartan equations (MC) existing in the theory of Lie groups, on the condition to change the sign of the structure constants involved because we have [α ρ , α σ ] = −c τ ρσ α τ . Writing these equations in the form of the exterior system dω τ = c τ ρσ ω ρ ∧ ω σ and closing this system by applying once more the exterior derivative d, we obtain the quadratic IC: and is a Lie group with a maximum number of n(n + 1)/2 parameters. A special metric could be the Euclidean metric when n = 1, 2, 3 as in elasticity theory or the Minkowski metric when n = 4 as in special relativity ( [12]). The first order Medolaghi equations: are also called classical Killing equations for historical reasons. The main problem is that this system is not involutive unless we prolong it to order two by differentiating once the equations. For such a purpose, introducing ω −1 = (ω ij ) as usual, we may define the Christoffel symbols: This is a new geometric object of order 2 providing the Levi-Civita isomorphism j 1 (ω) = (ω, ∂ω) ≃ (ω, γ) of affine bundles and allowing to obtain the second order Medolaghi equations: 0 Surprisingly, the following expression called Riemann tensor: is still a first order geometric object and even a 4-tensor with n 2 (n 2 − 1)/12 independent components satisfying the purely algebraic relations : , ω rl ρ l kij + ω kr ρ r lij = 0 Accordingly, the IC must express that the new first order equations R k lij ≡ (L(ξ)ρ) k lij = 0 are only linear combinations of the previous ones and we get the Vessiot structure equations: with the only structure constant c describing the constant Riemannian curvature condition of Eisenhart ( [4], [16,p139]). One can proceed similarly for the conformal Killing system L(ξ)ω = A(x)ω and obtain that the Weyl tensor must vanish, without any structure constant involved ([16,p 141]). EXAMPLE 1.10: (Contact structure) We only treat the case dim(X) = 3 as the case dim(X) = 2p + 1 needs much more work ( [15,p684]). Let us consider the so-called contact 1-form α = dx 1 − x 3 dx 2 and consider the Lie pseudogroup Γ ⊂ aut(X) of (local) transformations preserving α up to a function factor, that is Γ = {f ∈ aut(X)|j 1 (f ) −1 (α) = ρα} where again j q (f ) is a symbolic way for writing out the derivatives of f up to order q and α transforms like a 1-covariant tensor. It may be tempting to look for a kind of "object " the invariance of which should characterize Γ. Introducing the exterior derivative dα = dx 2 ∧ dx 3 as a 2-form, we obtain the volume As it is well known that the exterior derivative commutes with any diffeomorphism, we obtain sucessively: As the volume 3-form α ∧ dα transforms through a division by the Jacobian determinant ∆ = , the desired object is thus no longer a 1-form but a 1-form density ω = (ω 1 , ω 2 , ω 3 ) transforming like a 1-form but up to a division by the square root of the Jacobian determinant. It follows that the infinitesimal contact transformations are vector fields ξ ∈ T = T (X) the tangent bundle of X, satisfying the 3 so-called first order Medolaghi equations: When ω = (1, −x 3 , 0), we obtain the special involutive system: For an arbitrary ω, we may ask about the differential conditions on ω such that all the equations of order r + 1 are only obtained by differentiating r times the first order equations, exactly like in the special situation just considered where the system is involutive. We notice that, in a symbolic way, ω ∧ dω is now a scalar c(x) providing the zero order equation ξ r ∂ r c(x) = 0 and the condition is c(x) = c = cst. The integrability condition (IC) is the Vessiot structure equation: involving the only structure constant c.
FIRST FUNDAMENTAL RESULT : Comparing the various Vessiot structure equations containing structure constants that we have just presented and that we recall below in a symbolic way, we notice that these structure constants are absolutely on equal footing though they have in general nothing to do with any Lie algebra.
Accordingly, the fact that the ones appearing in the MC equations are related to a Lie algebra is a coincidence and the Cartan structure equations have nothing to do with the Vessiot structure equations. Also, as their factors are either constant, linear or quadratic, any identification of the quadratic terms appearing in the Riemann tensor with the quadratic terms appearing in the MC equations is definitively not correct ( [22]). We also understand why the torsion is automatically combined with curvature in the Cartan structure equations but totally absent from the Vessiot structure equations, even though the underlying group (translations + rotations) is the same.
HISTORICAL REMARK 1.12: Despite the prophetic comments of the italian mathematician Ugo Amaldi in 1909 ([16,p46-52]), it has been a pity that Cartan deliberately ignored the work of Vessiot at the beginning of the last century and that the things did not improve afterwards in the eighties with Spencer and coworkers (Compare MR 720863 (85m:12004) and MR 954613 (90e:58166)).

SECOND PART : THE JANET AND SPENCER SEQUENCES
Let µ = (µ 1 , ..., µ n ) be a multi-index with length |µ| = µ 1 + ... + µ n , class i if µ 1 = ... = µ i−1 = 0, µ i = 0 and µ + 1 i = (µ 1 , ..., µ i−1 , µ i + 1, µ i+1 , ..., µ n ). We set y q = {y k µ |1 ≤ k ≤ m, 0 ≤ |µ| ≤ q} with y k µ = y k when |µ| = 0. If E is a vector bundle over X with local coordinates (x, y) and J q (E) is the q-jet bundle of E with local coordinates (x, y q ), the Spencer operator just allows to distinguish a section ξ q from a section j q (ξ) by introducing a kind of "difference" through the operator D : ..) and more generally (Dξ q+1 ) k µ,i (x) = ∂ i ξ k µ (x)−ξ k µ+1i (x). Minus the restriction of D to the kernel S q+1 T * ⊗ E of the canonical projection π q+1 q : J q+1 (E) → J q (E) can be extended to the Spencer map δ : The kernel of D is made by sections such that ξ q+1 = j 1 (ξ q ) = j 2 (ξ q−1 ) = ... = j q+1 (ξ). Finally, if R q ⊂ J q (E) is a system of order q on E locally defined by linear is locally defined when r = 1 by the linear equations Φ τ (x, y q ) = 0, d i Φ τ (x, y q+1 ) ≡ a τ µ k (x)y k µ+1i + ∂ i a τ µ k (x)y k µ = 0 and has symbol g q+r = R q+r ∩ S q+r T * ⊗ E ⊂ J q+r (E) locally defined by a τ µ k (x)ξ k µ+ν = 0, | µ |= q, | ν |= r if one looks at the top order terms. If ξ q+1 ∈ R q+1 is over ξ q ∈ R q , differentiating the identity a τ µ k (x)ξ k µ (x) ≡ 0 with respect to x i and substracting the identity a τ µ k (x)ξ k µ+1i (x) + ∂ i a τ µ k (x)ξ k µ (x) ≡ 0, we obtain the identity a τ µ k (x)(∂ i ξ k µ (x) − ξ k µ+1i (x)) ≡ 0 and thus the restriction D : R q+1 → T * ⊗ R q . This first order operator induces, up to sign, the purely algebraic monomorphism 0 → g q+1 δ → T * ⊗ g q on the symbol level ( [17], [24). The Spencer operator has never been used in GR. DEFINITION 2.1: R q is said to be formally integrable (FI) when the restriction π q+r+1 q+r : R q+r+1 → R q+r is an epimorphism ∀r ≥ 0. In that case, the Spencer form R q+1 ⊂ J 1 (R q ) is a canonical equivalent formally integrable first order system on R q with no zero order equations. DEFINITION 2.2: R q is said to be involutive when it is formally integrable and the symbol g q is involutive, that is all the sequences ... δ → ∧ s T * ⊗ g q+r δ → ... are exact ∀0 ≤ s ≤ n, ∀r ≥ 0. Equivalently, using a linear change of local coordinates if necessary, we may successively solve the maximum number β n q , β n−1 q , ..., β 1 q of equations with respect to the leading or principal jet coordinates of strict order q and class n, n − 1, ..., 1. Then R q is involutive if R q+1 is obtained by only prolonging the β i q equations of class i with respect to d 1 , ..., d i for i = 1, ..., n. In that case, such a prolongation procedure allows to compute in a unique way the principal jets from the parametric other ones and may also be applied to nonlinear systems as well ( [6], [17]).
When R q is involutive, the linear differential operator D : . Each operator D r+1 : F r → F r+1 is first order involutive as it is induced by D : ∧ r T * ⊗ J q+1 (E) → ∧ r+1 T * ⊗ J q (E) : α ⊗ ξ q+1 → dα ⊗ ξ q + (−1) r α ∧ Dξ q+1 and generates the compatibility conditions (CC) of the preceding one. As the Janet sequence can be cut at any place, the numbering of the Janet bundles has nothing to do with that of the Poincaré sequence, contrary to what many people believe in GR. Similarly, we have the involutive first Spencer operator D 1 : C 0 = R q j1 → J 1 (R q ) → J 1 (R q )/R q+1 ≃ T * ⊗ R q /δ(g q+1 ) = C 1 of order one induced by D : R q+1 → T * ⊗ R q . Introducing the Spencer bundles C r = ∧ r T * ⊗ R q /δ(∧ r−1 T * ⊗ g q+1 ), the first order involutive (r + 1)-Spencer operator D r+1 : C r → C r+1 is induced by D : ∧ r+1 T * ⊗ R q+1 → ∧ r+1 T * ⊗ R q and we obtain the canonical linear Spencer sequence ([17, p 150]): as the Janet sequence for the first order involutive system R q+1 ⊂ J 1 (R q ). Introducing the other , the linear Spencer sequence is induced by the linear hybrid sequence: which is at the same time the Janet sequence for j q and the Spencer sequence for J q+1 (E) ⊂ J 1 (J q (E)) ( [17, p 153]). Such a sequence projects onto the Janet sequence and we have the following commutative diagram with exact columns: In this diagram, only depending on the linear differential operator D = Φ • j q , the epimorhisms Φ r : C r (E) → F r for 0 ≤ r ≤ n are induced by the canonical projection Φ = Φ 0 : C 0 (E) = J q (E) → J q (E)/R q = F 0 if we start with the knowledge of R q ⊂ J q (E) or from the knowledge of an epimorphism Φ : J q (E) → F 0 if we set R q = ker(Φ). In the theory of Lie equations coinsidered, E = T , R q ⊂ J q (T ) is a transitive involutive system of infinitesimal Lie equations of order q and the corresponding operator D is a Lie operator. As an exercise, we invite the reader to draw this diagram in the affine and projective 1-dimensional cases. EXAMPLE 2.3 : If we restrict our study to the group of isometries of the euclidean metric ω in dimension n ≥ 2, exhibiting the Janet and the Spencer sequences is not easy at all, even when n = 2, because the corresponding Killing operator Dξ = L(ξ)ω = Ω ∈ S 2 T * , involving the Lie derivative L and providing twice the so-called infinitesimal deformation tensor ǫ of continuum mechanics, is not involutive. In order to overcome this problem, one must differentiate once by considering also the Christoffel symbols γ and add the operator L(ξ)γ = Γ ∈ S 2 T * ⊗ T . Now, one can prove that the Spencer sequence for Lie groups of transformations is locally isomorphic to the tensor product of the Poincaré sequence by the Lie algebra of the underlying Lie group. Hence, if two Lie groups G ′ ⊂ G act on X, it follows from the definition of the Janet and Spencer bundles that the Spencer sequence for G ′ is embedded into the Spencer sequence for G while the Janet sequence for G ′ projects onto the Janet sequence for G but the common differences are isomorphic to ∧ r T * ⊗ (G/G ′ ). This rather philosophical comment, namely to replace the Janet sequence by the Spencer sequence, must be considered as the crucial key for understanding the work of the brothers E. and F. Cosserat in 1909 ([3], [19], [21], [22]) or H. Weyl in 1918 ([22], [26]), the best picture being that of Janet and Spencer playing at see-saw. Indeed, when n = 2, one has 3 parameters (2 translations + 1 rotation) and the following commutative diagram which only depends on the left commutative square: In this diagram, there is no way to compare D 1 (curvature alone as in Vessiot) with D 2 (curvature + torsion as in Cartan).
For proving that the adjoint of D 1 provides the Cosserat equations which can be parametrized by the adjoint of D 2 , we may lower the upper indices by means of the constant euclidean metric and look for the factors of ξ 1 , ξ 2 and ξ 1,2 = −ξ 2,1 in the integration by parts of the sum: in order to obtain: Finally, we get the nontrivial first order parametrization σ 11 = ∂ 2 φ 1 , by means of the three arbitrary functions φ 1 , φ 2 , φ 3 , in a coherent way with the Airy second order parametrization obtained if we set φ 1 = ∂ 2 φ, φ 2 = ∂ 1 φ, φ 3 = −φ when µ 1 = 0, µ 2 = 0 as we shall see in the third part.
The link between the FI of R q and the CC of D is expressed by the following diagram that may be used inductively: The " snake theorem " ( [17], [23]) then provides the long exact connecting sequence: If we apply such a diagram to first order Lie equations with no zero or first order CC, we have q = 1, E = T and we may apply the Spencer δ-map to the top row with r = 2 in order to get the commutative diagram: with exact rows and exact columns but the first that may not be exact at ∧ 2 T * ⊗ g 1 . We shall denote by B 2 (g 1 ) the coboundary as the image of the central δ, by Z 2 (g 1 ) the cocycle as the kernel of the lower δ and by H 2 (g 1 ) the Spencer δ-cohomology at ∧ 2 T * ⊗ g 1 as the quotient.
Comparing the classical and conformal Killing systems by using the inclusions R 1 ⊂R 1 ⇒ g 1 ⊂ g 1 , we finally obtain the following commutative and exact diagram where a diagonal chase allows to identify Ricci with S 2 T * ⊂ T * ⊗ T * ≃ T * ⊗ĝ 2 and it split the right column ( [22]): The Ricci tensor only depends on the "difference" existing between the clasical Killing system and the conformal Killing system, namely the n second order jets (elations once more). The Ricci tensor, thus obtained without contracting the indices as usual, may be embedded in the image of the Spencer operator made by 1-forms with value in 1forms that we have already exhibited for describing EM. It follows that the foundations of both EM and GR are not coherent with jet theory and must therefore be revisited within this new framework.
THIRD PART: ALGEBRAIC ANALYSIS: EXAMPLE 3.1: Let a rigid bar be able to slide along an horizontal axis with reference position x and attach two pendula, one at each end, with lengths l 1 and l 2 , having small angles θ 1 and θ 2 with respect to the vertical. If we project Newton law with gravity g on the perpendicular to each pendulum in order to eliminate the tension of the threads and denote the time derivative with a dot, we get the two equations: x + l 1θ1 + gθ 1 = 0,ẍ + l 2θ2 + gθ 2 = 0 As an experimental fact, starting from an arbitrary movement of the pendula, we can stop them if and only if l 1 = l 2 and we say that the system is controllable.
More generally, we can bring the OD equations describing the behaviour of a mechanical or electrical system to the Kalman formẏ = Ay+bu with input u = (u 1 , ..., u p ) and output y = (y 1 , ..., y m ). We say that the system is controllable if, for any given y(0), y(T ), T < ∞, one can find u(t) such that a coherent trajectory y(t) may be found. In 1963, R.E. Kalman discovered that the system is controllable if and only if rk(B, AB, ..., A m−1 B) = m. Surprisingly, such a functional definition admits a formal test which is only valid for Kalman type systems with constant coefficients and is thus far from being intrinsic. In the PD case, the Spencer form will replace the Kalman form. EXAMPLE 3.2:ẏ −u = 0 ⇒ y(t) − u(t) = c = cst ⇒ u(T ) − u(0) = y(T ) − y(0) can always be achieved and the system is thus controllable in the sense of the definition but z = y − u ⇒ż = 0 is not controllable in the sense of the test. EXAMPLE 3.3:ẏ 1 − a(t)y 2 −ẏ 3 = 0, y 1 −ẏ 2 +ẏ 3 = 0. Any way to bring this system to Kalman form provides the controllability condition a(a − 1) = 0 if a = cst but nothing can be said if a = a(t). Also, getting y 1 from the second equation and substituting in the first, we get the second order OD equationÿ 2 −ÿ 3 −ẏ 3 − a(t)y 2 = 0 for which nothing can be said at first sight.
PROBLEM 1: Is a SYSTEM of OD or PD equations " controllable " (answer must be YES or NO) and how can we define controllability ?. Now, if a differential operator ξ D −→ η is given, a direct problem is to find (generating) compatibility conditions (CC) as an operator η D1 −→ ζ such that Dξ = η ⇒ D 1 η = 0. Conversely, the inverse problem will be to find θ D−1 −→ ξ such that D generates the CC of D −1 and we shall say that D is parametrized by D −1 . Of course, solving the direct problem (Janet, Spencer) is necessary for solving the inverse problem. EXAMPLE 3.4: When n = 2, the Cauchy equations for the stress in continuum mechanics are ∂ 1 σ 11 + ∂ 2 σ 21 = 0, ∂ 1 σ 12 + ∂ 2 σ 22 = 0 with σ 12 = σ 21 . Their parametrization σ 11 = ∂ 22 φ, σ 12 = σ 21 = −∂ 12 φ, σ 22 = ∂ 11 φ has been discovered by Airy in 1862 and φ is called the Airy function. When n = 3, Maxwell and Morera discovered a similar parametrization with 3 potentials (exercise). EXAMPLE 3.5: When n = 4, the Maxwell equations dF = 0 where F ∈ ∧ 2 T * is the EM field are parametrized by dA = F where A ∈ T * is the 4-potential. The second set of Maxwell equations can also be parametrized by the so-called pseudopotential which is a pseudovector density (exercise). EXAMPLE 3.5: If n = 4, ω is the Minkowski metric and φ = GM/r is the gravitational potential, then φ/c 2 ≪ 1 and a perturbation Ω of ω may satisfy in vacuum the 10 second order Einstein equations for the 10 Ω: The parametrizing challenge has been proposed in 1970 by J. Wheeler for 1000 $ and solved negatively in 1995 by the author who only received 1 $. PROBLEM 2: Is an OPERATOR parametrizable (answer must be YES or NO) and how can we find a parametrization ?.
Let A be a unitary ring, that is 1, a, b ∈ A ⇒ a + b, ab ∈ A, 1a = a and even an integral domain, that is ab = 0 ⇒ a = 0 or b = 0. We say that M = A M is a left module over A if x, y ∈ M ⇒ ax, x + y ∈ M, ∀a ∈ A and we denote by hom A (M, N ) the set of morphisms f : M → N such that f (ax) = af (x). PROBLEM 3: Is a MODULE M torsion-free, that is t(M ) = 0 (answer must be YES or NO) and how can we test such a property ?.
In the remaining of this paper we shall prove that the three problems are indeed identical and that only the solution of the third will provide the solution of the two others.
Let Q ⊂ K be a differential field, that is a field (a ∈ K ⇒ 1/a ∈ K) with n commuting Using an implicit summation on multiindices, we may introduce the (noncommutative) ring of differential operators D = K[d 1 , ..., d n ] = K[d] with elements P = a µ d µ such that µ < ∞ and d i a = ad i + ∂ i a. Now, if we introduce differential indeterminates y = (y 1 , ..., y m ), we may extend d i y k µ = y k µ+1i to Φ τ ≡ a τ µ k y k µ di −→ d i Φ τ ≡ a τ µ k y k µ+1i + ∂ i a τ µ k y k µ for τ = 1, ..., p. Therefore, setting Dy 1 + ... + dy m = Dy ≃ D m , we obtain by residue the differential module or D-module M = Dy/DΦ. Introducing the two free differential modules F 0 ≃ D m , F 1 ≃ D p , we obtain equivalently the free presentation F 1 D → F 0 → M → 0. More generally, introducing the successive CC as in the preceding section, we may finally obtain the free resolution of M , namely the exact sequence ...

D2
trick " is to let D act on the left on column vectors in the operator case and on the right on row vectors in the module case. Homological algebra has been created for finding intrinsic properties of modules not depending on their presentation or even on their resolution. EXAMPLE 3.7: In order to understand that different presentations may nevertheless provide isomorphic modules, let us consider the linear inhomogeneous system P y ≡ d 22 y = u, Q ≡ d 12 y−y = v with K = Q. Differentiating twice, we get y = d 11 u − d 12 v − v and the two fourth order CC: However, as P Q = QP , we also get the CC C ≡ d 12 u − d 22 v − u = 0 and the two resolutions: where we have identified the two differential modules involved on the right because: We now exhibit another approach by defining the formal adjoint of an operartor P and an operator matrix D:  ∧ n T * ⊗ E * ad(D) ←− ∧ n T * ⊗ F * where E * is obtained from E by inverting the transition matrix and EM provides a fine example of such a procedure ( [12]). Now, with operational notations, let us consider the two differential sequences: EXAMPLE 3.10: With ∂ 22 ξ = η 2 , ∂ 12 ξ = η 1 for D, we get ∂ 1 η 2 − ∂ 2 η 1 = ζ for D 1 . Then ad(D 1 ) is defined by µ 2 = −∂ 1 λ, µ 1 = ∂ 2 λ while ad(D) is defined by ν = ∂ 12 µ 1 + ∂ 22 µ 2 but the for the stress tensor density ( [16], [26] Accordingly, the second order Airy parametrization is nothing else than the adjoint of the only Riemann CC involved, namely ∂ 11 ǫ 22 + ∂ 22 ǫ 11 − 2∂ 12 ǫ 12 = 0 which is the linearization of the Riemann tensor of Example 1.9 . EXERCISE 3.20: Prove thatÿ 2 −ÿ 3 −ẏ 3 − a(t)y 2 = 0 is controllable if and only ifȧ + a 2 − a = 0 (Riccati) and find a parametrization.